How do I get an object to stay put when keyframing child of constraint influence?
You animated the "influence" on your "Child of" constraint. On frame 0, it was set to Influence of 1.00, but then its influence goes all the way to 0.00 at frame 1. 
This means that you must have accidentally animated the influence from 1.00 to 0.00 in the animation. So, just reset the Influence value back to 1.00 at frame 1.